How to make half of a square circular
Add a Cylinder ShiftA > Mesh > Cylinder, set the Cap Fill Type to Nothing.
Select half of it and delete with X or Delete 
Select the four vertices along the edge
Extrude to make the long sides

Fill the back, top and bottom to get this.



Note the top and bottom are Ngons, witch in many cases are undesirable. To get a quads all over the top use grid fill.